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Concerning the definition of symplectic K-theory: there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one …

For commutative rings, all the K-theory computations I am aware of are discussed in Chapter VI of Weibel's $K$-book (actually with the exception of finite fields). Function rings: I would say that f …

This is going to be a slightly extended explanation, I apologize. The short version is basically that little is actually known (and even that is hard to prove), but conjecturally everything permitted …

I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion. The trouble starts with $K_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pre …

The category of flat vector bundles is equivalent to the category of local systems, see for instance this MO-question, which in turn are equivalent to representations of the fundamental group, see thi …

The previous answer contained a major error/misconception, and I apologize for the dealy in correcting it. The answer to the question is "yes" locally in the Zariski topology but "no" globally. Comp …

Here are some known computations for infinite discrete groups. Basically, most of these proceed by computing the equivariant K-homology of the classifying space of proper actions and deduce the comput …

Let me flesh out my comment, and give some more details. I will denote $C_p$ the cyclic group of order $p$. First, the structure of the group ring $\mathbb{Z}[C_p]$ is a little more complicated. If …

There are two papers that could be interesting to you. First, there is a paper of Kevin Hutchinson: K. Hutchinson. On the low-dimensional homology of ${\rm SL}_2(k[t,t^{-1}])$. J. Algebra 425 (2015) …

We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology): $$ 0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}_n\to \mathbf{K}^{\rm M}_n\to 0, …

The degrees of the attaching maps (and hence the integral chain complex) for the real Grassmannians have been determined in L. Casian and Y. Kodama. On the cohomology of real Grassmann manifolds. a …

I would like to add a couple of references to the ones provided in Timo Keller's comment. First of all, the conjecture is not correctly stated. According to the conjectures, the rank of the curve (w …

The Atiyah-Hirzebruch spectral sequence does have a multiplicative structure, and I think this can be used to determine the multiplication on K-theory. From the paper of Braun, it follows that the spe …

First, some general remarks on the situation for $R$ an arbitrary commutative ring. Since ${\rm diag}(-1,-1)$ acts by multiplication by $-1$ on $R^{\oplus 2}$, the homology groups ${\rm H}_i({\rm SL}_ …

Yes, the natural multiplication morphisms induce a morphism of Gersten complexes from Milnor to Quillen K-theory. The basic points are made in the paper M. Rost. "Chow groups with coefficients", Do …